Page 714 #1-10 ANSWERS

1. Page 714 #1-10 ANSWERS

3. Student Learning Goal Chart Lesson Reflection for Chapter 14 Section 7

4. Pre-Algebra Learning Goal Students will understand collecting, displaying, & analyzing data.

5. Students will understand collecting, displaying & analyzing data by being able to do the following: • Learn to identify populations & recognize biased samples (4-1) • Learn to organize data in tables and stem-and-leaf plots (4-2) • Learn to find Euler circuits (14-6) • Learn to find and use Hamiltonian circuits (14-7)

6. Today’s Learning Goal Assignment Learn to find and use Hamiltonian circuits

7. Pre-Algebra HW Page 718 #1-6

8. Hamiltonian Circuits 14-7 Warm Up Problem of the Day Lesson Presentation Pre-Algebra

9. Hamiltonian Circuits 14-7 A D B C Warm Up 1.Determine whether the graph can be traversed (traveled) through an Euler circuit. Explain. Yes; all vertices have even degrees. Pre-Algebra

10. Hamiltonian Circuits 14-7 (3a3 – a2 – 3) + = 4a3 + a2 Problem of the Day Find the missing polynomial that makes the equation true. a3 + 2a2 + 3 Pre-Algebra Pre-Algebra

11. Hamiltonian Circuits 14-7 Today’s Learning Goal Assignment Learn to find and use Hamiltonian circuits. Pre-Algebra Pre-Algebra

12. Hamiltonian Circuits 14-7 Insert Lesson Title Here Vocabulary Hamiltonian circuit Pre-Algebra Pre-Algebra

13. Hamiltonian Circuits 14-7 Reading Math Note that the lengths shown are distances, not side lengths of geometric figures. Roger and his friends are planning a trip to four Major League baseball parks. They will start their trip at Busch Stadium, in St. Louis, visit each stadium once, and then return to Busch Stadium. Use the graph on the next slide to find a path that Roger and his friends can travel. Pre-Algebra Pre-Algebra

14. Hamiltonian Circuits 14-7 Wrigley Field (Chicago) 269 mi 10 mi Busch Stadium (St. Louis) 99 mi 374 mi 301 mi Comisky Park (Chicago) 92 mi Miller Park (Milwaukee) A Hamiltonian circuit is a path that ends at the beginning vertex and passes through each of the other vertices in the graph exactly once. In a Hamiltonian circuit, the path is not required to traverse each edge. Pre-Algebra Pre-Algebra

15. Hamiltonian Circuits 14-7 Additional Example 1: Finding Hamiltonian Circuits Find a Hamiltonian circuit in the graph. Start/ Finish Visit each vertex once, but do not necessarily traverse each edge. Pre-Algebra Pre-Algebra

16. Hamiltonian Circuits 14-7 Insert Lesson Title Here Try This: Example 1 Find a Hamiltonian circuit in the graph. Start/ Finish Visit each vertex once, but do not necessarily traverse each edge. Pre-Algebra Pre-Algebra

17. Hamiltonian Circuits 14-7 Additional Example 2: Problem Solving Application Use the information in the graph to determine the shortest Hamiltonian circuit beginning at Boston. Boston Cleveland 522 mi 190 mi 408 mi 360 mi New York 1,086 mi 1088 mi 170 mi Miami Baltimore 955 mi Pre-Algebra Pre-Algebra

18. Hamiltonian Circuits 14-7 1 3 Solve Choose a Strategy Analyze the Problem 2 Additional Example 2 Continued Find the shortest path between the cities. In the graph the vertices represent different cities. Find all the Hamiltonian circuits beginning and ending at Boston. Find the length of each path. The first letter of each city’s name is used to represent the city. Pre-Algebra Pre-Algebra

19. Hamiltonian Circuits 14-7 3 Solve Additional Example 2 Continued B 190 N 170 B 955 M 1086 C 522 B = 2923 B 190 N 408 C 1086 M 955 B 360 B = 2999 B 522 C 1086 M 1088 N 170 B 360 B = 3226 B 360 B 170 N 1088 M 1086 C 522 B = 3196 B 522 C 1086 M 955 B 170 N 190 B = 2923 B 360 B 955 M 1088 N 408 C 522 B = 3333 B 360 B 955 M 1086 C 408 N 190 B = 2999 B 522 C 408 N 1088 M 955 B 360 B = 3333 The shortest circuit is BN B MCB or BCM B N B. Pre-Algebra Pre-Algebra

20. Hamiltonian Circuits 14-7 4 Additional Example 2 Continued Look Back Be sure that you have all possible paths. There are 8 paths that meet the Hamiltonian criteria of only passing through each vertex once. Pre-Algebra Pre-Algebra

21. Hamiltonian Circuits 14-7 Try This: Example 2 Use the information in the graph to determine the shortest Hamiltonian circuit beginning and ending at B. B 4 in. 6 in. 2 in. 4 in. E C A 3 in. 7 in. 5 in. D Pre-Algebra Pre-Algebra

22. Hamiltonian Circuits 14-7 1 Choose a Strategy Analyze the Problem 2 Insert Lesson Title Here Try This: Example 2 Find the shortest path between locations. In the graph the vertices represent different locations. Find all the Hamiltonian circuits beginning and ending at B. Pre-Algebra Pre-Algebra

23. Hamiltonian Circuits 14-7 3 Solve Try This: Example 2 Find the length of each path. The vertex locations are represented by letters. B 6 A 4 E 3 D 7 C 4 B = 24 in. B 4 C 2 E 3 D 5 A 6 B = 20 in. B 6 A 5 D 3 E 2 C 4 B = 20 in. B 4 C7 D 3 E 4 A 6 B = 24 in. The shortest circuit is BCED A B or B A DECB. Pre-Algebra Pre-Algebra

24. Hamiltonian Circuits 14-7 Look Back 4 Try This: Example 2 Be sure that you have all possible paths. There are 4 paths that meet the Hamiltonian criteria of only passing through each vertex once and beginning and ending at B. Pre-Algebra Pre-Algebra

25. Hamiltonian Circuits 14-7 Insert Lesson Title Here Lesson Quiz Use the graph for problems 1-2. A 12 in. 4 in. 6 in. 10 in. D C 4 in 13 in. B 1. Find a Hamiltonian circuit in the graph. Possible answer: A-B-C-D-A 2. Determine the shortest Hamiltonian circuit beginning and ending at A, and find its length. A-B-D-C-A or A-C-D-B-A; 32 in. Pre-Algebra Pre-Algebra

26. Page 718 #1-6 ANSWERS